Polynomials and Multiple Derivatives

In this applet you can move the seven red control points up and down. A 6th degree polynomial
is fitted to these points, so you change the polynomial by moving the control points.

Explore

If you take the derivative of a 6th degree polynomial, must it always be
the case that you get a 5th degree polynomial? Are there any exceptions?

Looking at the graph of a polynomial, how can you tell, in general, what
the degree of the polynomial is?

When the slider shows `d = 0`, the original 6th degree polynomial is displayed.
Higher values of `d` take higher derivatives. Play with the slider
and confirm that the derivatives of the polynomial behave the way you expect.